The subject matter of signal sampling is widely known in the prior art. Generally, it relates to digital signal processing and has high relevance in a variety of fields, such as communication, electronics, medicine, electro-optics, and many others. For example, in radio communication, sampling a signal and obtaining sufficient signal attenuation, while demodulating the desired signal from radio frequencies as close as possible to the baseband, is one of the main tasks. According to the commonly known Nyquist-Shannon sampling theorem, which is well known in the field of information theory, and in particular, in the field of digital signal processing and telecommunications, an analog signal that has been sampled can be fully reconstructed from the samples if the sampling frequency FS exceeds 2B samples per second (2B is a Nyquist rate that is the minimum sampling rate required to avoid aliasing), where B is the bandwidth of the original signal, i.e. FS>2B or FS/2>B (half of the sampling rate is larger than the signal bandwidth). However, the above theorem is valid when the signal frequency range does not contain whole multiples or half-multiples of the sampling rate (sampling frequency).
It should be noted that signals that are used in many applications are, in many cases, band limited to a predefined frequency interval, and thus these signals are called bandpass signals. A uniform sampling theorem for bandpass signal is known from the prior art, and its analysis is usually based on the time frequency equivalence. Thus, for example, A. W. Kohlenberg proposed the second order sampling for a bandpass signal (in the article titled “Exact interpolation of band-limited functions”, published in the journal of Applied Physics, in 1953, issue 24(12), pages 1432-1436), which is considered to be the simplest case of non-uniform sampling where two uniform sampling sequences are interleaved. Second order sampling allows the theoretical minimal sampling rate of two-times bandwidth, in the form of an average rate, to be applied independent of the band position. In second order sampling, when the delay τ between two or more samplers is properly predefined, the signal can be fully reconstructed (e.g., by performing signal interpolation) even when the signal frequency range contains whole multiples or half-multiples of the sampling frequency.
FIG. 1A schematically illustrates a conventional interpolation system 100 of second order sampling, according to the prior art. In FIG. 1A, the input signal X(t) (t is a time parameter) passes through two Analog-to-Digital (A/D) converters 105′ and 105″ with a predefined time delay τ between them. Then, the converted signals X1(l) and X2(l) are inputted into interpolation filters 110′ and 110″, respectively, for performing signal interpolation, which includes digital to analog conversion. After that, the resulting interpolated signals are summed together, giving rise to the output signal Y(l), and in turn Y(t).
It should be noted that second order sampling and its limitations are well-known in the prior art, and this issue is discussed in the literature. For example, R. G. Vaughan et al., in the article titled “The Theory of Bandpass Sampling” published in the “IEEE Transactions on Signal Processing” journal (volume 39, number 2, pp. 1973-1984, September 1991), discusses sampling of bandpass signals with respect to band position, noise considerations, and parameter sensitivity, presenting acceptable and unacceptable sample rates with specific discussion of the practical rates which are non-minimum. According to Vaughan et al., the construction of a bandpass signal from second-order samples depends on sampling factors and the relative delay between the uniform sampling streams. For another example, M. Valkama et al., in the article titled “A Novel Image Rejection Architecture for Quadrature Radio Receivers” published in the “IEEE Transactions on Circuits and Systems” journal (volume 51, number 2, pp. 61-68, February 2004), presents a novel structure for obtaining an image-free baseband observation of the received bandpass signal by utilizing I/Q (Inphase/Quadrature) signal processing. The phase difference between I and Q branches is approximated by a relative time delay of one quarter of the carrier cycle. Also, Valkama et al. presents and analyzes an analog delay processing based model, and then determines the obtainable image rejection of the delay processing. In addition, Valkama et al. in another article titled “Second-Order Sampling of Wideband Signals”, published in the “IEEE International Symposium on Circuits and Systems” journal (volume 2, pp. 801-804, May 2001), discusses and analyzes the second-order sampling based digital demodulation technique. According to Valkama et al., the modest image rejection of the basic second-order sampling scheme is improved to provide sufficient demodulation performance also for wideband receivers. Further, for example, H. Yong et al. in the article titled “Second-Order Based Fast Recovery of Bandpass Signals”, published in the “International Conference on Signal Processing Proceedings” journal (volume 1, pp. 7-10, 1998), discusses fast recovery and frequency-differencing of real bandpass signals based on second-order sampling. According to H. Yong et al., by using second-order sampling, the sampling rate can be lowered to the bandwidth. Although the spectrum of the two interleaved sampling streams are aliasing, it is possible to reconstruct the original or frequency-differencing bandpass signal.
Further, it should be noted that the conventional complex signal processing is also used in processing schemes where an input signal is bandpass in its origin, and is to be processed in a lowpass form. This normally requires two-channel processing in quadrature channels to remove an ambiguity as to whether a signal is higher or lower than the bandpass center frequency. The complex signal processing can be extended to the digital signal processing field, and the processed signal can be first mixed to zero-center frequency in two quadrature channels, then filtered to remove the high frequency mixing products, and after that digitized by a number of A/D (Analog-to-Digital) converters.
According to the prior art, FIG. 1B schematically illustrates a conventional complex sampling system 160, in which an input signal is sampled in two sampling channels 150′ and 150″, while shifting the phase by ninety degrees. At the output of such a system, a complex signal is obtained, said signal having a real part Re{X(l)} and an imaginary part Im{X(l)}, wherein parameter l represents a series of discrete values. Filters 151, 152′ and 152″ are used to filter the undesired frequency range (in a time domain) of input signals X(t), X1′(t) and X2′(t), respectively.
U.S. Pat. No. 5,099,194 discloses an approach to extending the frequency range uses non-uniform sampling to gain the advantages of a high sampling rate with only a modest increase in the number of samples. Two sets of uniform samples with slightly different sampling frequency are used. Each set of samples is Fourier transformed independently and the frequency of the lowest aliases determined. It is shown that knowledge of these two alias frequencies permits unambiguous determination of the signal frequency over a range far exceeding the Nyquist frequency, except at a discrete set of points.
U.S. Pat. No. 5,099,243 presents a technique for extending the frequency range which employs in-phase and quadrature components of the signal coupled with non-uniform sampling to gain the advantages of a high sampling rate with only a small increase in the number of samples. By shifting the phase of the local oscillator by 90 degrees, a quadrature IF signal can be generated. Both in-phase and quadrature components are sampled and the samples are combined to form a complex signal. When this signal is transformed, only one alias is obtained per periodic repetition and the effective Nyquist frequency is doubled. Two sets of complex samples are then used with the slightly different sampling frequency. Each set is independently Fourier transformed and the frequency of the lowest aliases permits unambiguous determination of the signal frequency over a range far exceeding the Nyquist frequency.
U.S. Pat. No. 5,109,188 teaches a technique for extending the frequency range which employs a power divider having two outputs, one output being supplied to a first Analog-to-Digital (A/D) converter, and the other output being supplied via a delay device to a second A/D converter. A processor receives the outputs of the two A/D converters. In operation, the input signal is subjected to a known delay and both original and delayed signals are sampled simultaneously. Both sampled signals are Fourier transformed and the phase and amplitudes calculated. The phase difference between the original and delayed signals is also calculated, and an approximation to the true frequency for each peak observed in the amplitude spectrum is estimated.
Based on the above observations, there is a continuous need in the art to provide a method and system configured to perform complex sampling of signals by using two or more sampling channels (second-order sampling or higher) and enabling operating with a signal bandwidth that can be equal to the sampling frequency (or to higher multiples of the sampling frequency). In addition, there is a need in the art to provide a method and system for performing signal processing by using second order (or higher order) sampling, in a frequency domain, without considering whether the signal frequency range contains whole multiples or half-multiples of the sampling frequency. Further, there is a continuous need in the prior art to enable calculating corresponding time delays between the two or more sampling channels in a relatively accurate manner.